>>But when you create a formal system, don't you still have to remain>>within the boundaries of a meta-system ?

>Not in any *formal* meta-system, surely: I reason informally until I have>created a formal system with the desired properties. Then I reason>semi-formally within that system to check that it has the desired>properties, and to see what other cool things it spits out. But often>the most fun part, and the really hard work, comes before everything is>formal.

Eventually you will have to build all the formal bridges, except maybe youraxioms. Of course, there's a lot of thought that can be done outside theformal frame; but at some time all must be tied down within a meta-system.I'm not disagreeing that one can reason outside the formal system, butthat one can't have results outside it.

Also, you're talking at a level far beyond that which I was addressing. When doing mathematical research, one probably has to use whatever toolsone has at his/her disposal, but learning it is another story. And whenwe're talking about teaching students, do we want to emphasize informalthought ? I'm not so sure about this, my computer science bias tellsme the opposite, if nothing else because being "informal" inside acomputer program quickly leads to chaos: it's too easy to subvert therules of intuition and insight when we posess the capacity of establishingthem to begin with.

>I'm no pure logician, but I certainly pop out of the formal frame >when I am sniffing my way towards interesting formalisms. Call me an >impure logician if you like; I suppose my work on category theory is a>form of logic highly contaminated by applications to mathematical>physics.

Logic itself is your frame, or isn't it ? I'm not at that level, but Ihave difficulty seeing any kind of mathematics that doesn't anchoritself in a logical frame. I know that category theory people, justlike lambda calculus people, like to see category theory as a formalsystem par with set theory and others. But all three of them are stillbased on a logic system, or at least I do so far believe.

>What I'm doing is nothing new or surprising, by the way! Just think of>all the nonrigorous calculations Euler did to find nice formulas like>>1 + 1/4 + 1/9 + 1/16 + .... = pi^2/6>>He wasn't reasoning in any formal system. Indeed, the whole obsession>with formal systems came much later, peaking around the beginning of the 20th>century. Luckily, mathematical physicists today are still no slaves to>formalism. Think of how Ed Witten won the Fields Medal, the biggest>math prize of all! His reasoning was not formal, and in many cases we>still don't know how to make it formal --- but his discoveries were>wonderful.

I hear you, and I don't totally disagree. Still, at some point in time they must come down to earth and grind the nitty-gritty; it's not easy to put out some result without a proof, it becomes a conjecture rather.And what's a proof if it isn't contained within a formal system ? I'm not familiar with the work of Ed Witten, are his proofs "informal", in the sense that they don't fit within a logic system ?